New dimension bounds for αβ sets

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Abstract

In this paper we obtain new lower bounds for the upper box dimension of αβ sets. As a corollary of our main result, we show that if α is not a Liouville number and β is a Liouville number, then the upper box dimension of any αβ set is 1. We also use our dimension bounds to obtain new results on affine embeddings of self-similar sets.
Original languageEnglish
JournalJournal of Number Theory
Early online date31 May 2021
DOIs
Publication statusE-pub ahead of print - 31 May 2021

Keywords

  • αβ sets
  • Diophantine approximation
  • Self-similar sets

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